Question: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{p^3 + p^2 - 42p}{4p^2 + 44p + 112}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {p(p^2 + p - 42)} {4(p^2 + 11p + 28)} $ $ y = \dfrac{p}{4} \cdot \dfrac{p^2 + p - 42}{p^2 + 11p + 28} $ Next factor the numerator and denominator. $ y = \dfrac{p}{4} \cdot \dfrac{(p + 7)(p - 6)}{(p + 7)(p + 4)}$ Assuming $p \neq -7$ , we can cancel the $p + 7$ $ y = \dfrac{p}{4} \cdot \dfrac{p - 6}{p + 4}$ Therefore: $ y = \dfrac{ p(p - 6)}{ 4(p + 4)}$, $p \neq -7$